The results are as follows :(1)We determine the limit of the lowest eigenvalue of a Schr"odinger operator on a Wiener space under semi-classical limit. This is the case where the coefficient operator of the Dirichlet form is identity operator. Also we extend the result to the case where the coefficient operator is variable case and the case of a Schr"odinger operator on a path space over a compact Riemannian manifold in a recent preprint. The points are to use a unitary transformation by an approximate ground state function and rough path analysis.(2)We prove a weak Poincare inequality on a loop space over a compact Riemannian manifold by using the rough path analysis.(3)We prove a weak Poincare inequality on a loop space over a compact Riemannian manifold by using the rough path analysis.